How to use Web Push API inside an Intranet?

I'm working on a web application that is hosted completely inhouse.
The app should be able to send web push notifications to the users. The server is connected to the internet, the clients (desktop browsers) are not.
Security concerns prohibit to send any informations from the app to the internet.
What I have learned from examples like this is that always an external service like http://ift.tt/2cu385O or http://ift.tt/1Trj85V is involved when sending web push notifications.

Now my questions:
(How) Is it possible to implement an "intranet based" sending service i. e. not relying on external services as stated above?
Could the Web Push library for Node.js be a solution (I have no experience with node.js at all up to date)?
Desktop browsers in use are Chrome and Firefox but this can change in future.

Thx.